Questions and Answers
A homogeneous system of equations can be inconsistent.
False
If x is a nontrivial solution of Ax=0, then every entry in x is nonzero.
False
The effect of adding p to a vector is to move the vector in a direction parallel to p.
True
The equation Ax=b is homogeneous if the zero vector is a solution.
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If Ax=b is consistent, then the solution set of Ax=b is obtained by translating the solution set of Ax=0.
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The equation Ax=b is referred to as a vector equation.
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A vector b is a linear combination of the columns of a matrix A if and only if the equation Ax=b has at least one solution.
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The equation Ax=b is consistent if the augmented matrix Ab has a pivot position in every row.
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The first entry in the product Ax is a sum of products.
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If the columns of an m×n matrix A span ℝm, then the equation Ax=b is consistent for each b in ℝm.
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If A is an m×n matrix and if the equation Ax=b is inconsistent for some b in ℝm, then A cannot have a pivot position in every row.
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Study Notes
Homogeneous Systems
- A homogeneous system of equations, represented as Ax=0, always has at least one solution, specifically x=0.
- Such a system cannot be inconsistent as it guarantees the existence of the trivial solution.
Nontrivial Solutions
- A nontrivial solution of Ax=0 is a nonzero vector that satisfies the equation.
- This solution can contain zero entries; only not all entries can be zero.
Vector Addition
- Adding a vector p to another vector v moves v in a direction parallel to p.
- This applies equally in ℝ2 and ℝ3, shifting the vector along the line from p to the origin.
Homogeneous Equation Characteristics
- An equation Ax=b is classified as homogeneous if it can be written in the format Ax=0.
- If the zero vector is a solution, then b must equal Ax=A0=0, affirming the homogeneous nature.
Consistent Solutions and Translations
- If Ax=b is consistent, its solution set can be expressed as translating the solution set of the homogeneous equation Ax=0.
- Each solution can be represented as w=p+vh, where p is a specific solution and vh represents any solution to the homogeneous equation.
Matrix vs. Vector Equation
- The expression Ax=b is categorized as a matrix equation due to the matrix A involved.
- It is important to distinguish this from vector equations.
Linear Combinations
- A vector b is a linear combination of matrix A's columns if there exists at least one solution to Ax=b.
- This concept relates directly to expressing the equation in terms of the matrix's columns and vector components.
Consistency Criteria
- The statement "the equation Ax=b is consistent if the augmented matrix Ab has a pivot position in every row" is false.
- A pivot position could exist in the b column, affecting the overall consistency of the equation.
Entry Calculation in Products
- The first entry of the product Ax is calculated as the sum of products of respective entries in vector x and the corresponding entries from the first column of A.
Column Spanning Implications
- If matrix A's columns span ℝm, it ensures that Ax=b is consistent for all b in ℝm.
- This property guarantees the existence of solutions for every possible b vector.
Pivot Position and Inconsistency
- For an m×n matrix A, if Ax=b is inconsistent for some b in ℝm, then it cannot have a pivot position in every row.
- This reinforces the direct relationship between pivot positions and the existence of solutions to the system of equations.
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Description
Test your understanding of Linear Algebra concepts with this True or False flashcard quiz. Focus on key definitions and properties related to homogeneous systems of equations and their solutions.
